Question:
What is the minimal boolean expression for the function \(F(A, B, C, D) = \sum m(0, 2, 8, 10, 14) + d(5, 15)\)?
फलन \(F(A, B, C, D) = \sum m(0, 2, 8, 10, 14) + d(5, 15)\) के लिए न्यूनतम बूलियन व्यंजक क्या है?
Options:
- \(\overline{B}\ \overline{D} + A C D\)
- \(\overline{B}\ \overline{D} + A C\)
- \(\overline{A}\ \overline{D} + B C\)
- \(\overline{B}\ D + A C\)
- \(\overline{B}\ \overline{D} + A C D\)
- \(\overline{B}\ \overline{D} + A C\)
- \(\overline{A}\ \overline{D} + B C\)
- \(\overline{B}\ D + A C\)
Correct Answer: a)
Explanation:
K-map analysis on 4 variables \(A, B, C, D\):
– Minterms: \(m_0 (0000), m_2 (0010), m_8 (1000), m_{10} (1010), m_{14} (1110)\).
– Don’t care: \(d_5 (0101), d_{15} (1111)\).
1. The four corner cells \(m_0, m_2, m_8, m_{10}\) form a quad representing \(\overline{B}\ \overline{D}\).
2. To cover \(m_{14} (1110)\), we can combine \(m_{14}\) with don’t care \(d_{15} (1111)\) to form a pair: \(111- \implies A B C\). Or group with other cells? Let’s check: \(m_{10}, m_{14}\) forms a pair \(-010 \implies A C \overline{D}\). But if combined with \(d_{15} (1111)\) and \(m_{14} (1110)\), it is \(A C D\) or \(A C \overline{D}\). Option A gives \(\overline{B}\ \overline{D} + A C D\).
– Minterms: \(m_0 (0000), m_2 (0010), m_8 (1000), m_{10} (1010), m_{14} (1110)\).
– Don’t care: \(d_5 (0101), d_{15} (1111)\).
1. The four corner cells \(m_0, m_2, m_8, m_{10}\) form a quad representing \(\overline{B}\ \overline{D}\).
2. To cover \(m_{14} (1110)\), we can combine \(m_{14}\) with don’t care \(d_{15} (1111)\) to form a pair: \(111- \implies A B C\). Or group with other cells? Let’s check: \(m_{10}, m_{14}\) forms a pair \(-010 \implies A C \overline{D}\). But if combined with \(d_{15} (1111)\) and \(m_{14} (1110)\), it is \(A C D\) or \(A C \overline{D}\). Option A gives \(\overline{B}\ \overline{D} + A C D\).
K-मैप द्वारा चारों कोनों \(m_0, m_2, m_8, m_{10}\) से \(\overline{B}\ \overline{D}\) पद प्राप्त होता है, और शेष को कवर करने पर न्यूनतम व्यंजक प्राप्त होता है।